**Purpose**

To compute orthogonal transformation matrices Q and Z which reduce the regular pole pencil A-lambda*E of the descriptor system (A-lambda*E,B,C) to the generalized real Schur form with ordered generalized eigenvalues. The pair (A,E) is reduced to the form ( * * * * ) ( * * * * ) ( ) ( ) ( 0 A1 * * ) ( 0 E1 * * ) Q'*A*Z = ( ) , Q'*E*Z = ( ) , ( 0 0 A2 * ) ( 0 0 E2 * ) ( ) ( ) ( 0 0 0 * ) ( 0 0 0 * ) where the subpencil A1-lambda*E1 contains the eigenvalues which belong to a suitably defined domain of interest and the subpencil A2-lambda*E2 contains the eigenvalues which are outside of the domain of interest. If JOBAE = 'S', the pair (A,E) is assumed to be already in a generalized real Schur form and the reduction is performed only on the subpencil A12 - lambda*E12 defined by rows and columns NLOW to NSUP of A - lambda*E.

SUBROUTINE TG01PD( DICO, STDOM, JOBAE, COMPQ, COMPZ, N, M, P, $ NLOW, NSUP, ALPHA, A, LDA, E, LDE, B, LDB, $ C, LDC, Q, LDQ, Z, LDZ, NDIM, ALPHAR, ALPHAI, $ BETA, DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER COMPQ, COMPZ, DICO, JOBAE, STDOM INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M, N, $ NDIM, NLOW, NSUP, P DOUBLE PRECISION ALPHA C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*), $ BETA(*), C(LDC,*), DWORK(*), E(LDE,*), $ Q(LDQ,*), Z(LDZ,*)

**Mode Parameters**

DICO CHARACTER*1 Specifies the type of the descriptor system as follows: = 'C': continuous-time system; = 'D': discrete-time system. STDOM CHARACTER*1 Specifies whether the domain of interest is of stability type (left part of complex plane or inside of a circle) or of instability type (right part of complex plane or outside of a circle) as follows: = 'S': stability type domain; = 'U': instability type domain. JOBAE CHARACTER*1 Specifies the shape of the matrix pair (A,E) on entry as follows: = 'S': (A,E) is in a generalized real Schur form; = 'G': A and E are general square dense matrices. COMPQ CHARACTER*1 = 'I': Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = 'U': Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned. This option can not be used when JOBAE = 'G'. COMPZ CHARACTER*1 = 'I': Z is initialized to the unit matrix, and the orthogonal matrix Z is returned; = 'U': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned. This option can not be used when JOBAE = 'G'.

N (input) INTEGER The number of rows of the matrix B, the number of columns of the matrix C, and the order of the square matrices A and E. N >= 0. M (input) INTEGER The number of columns of the matrix B. M >= 0. P (input) INTEGER The number of rows of the matrix C. P >= 0. NLOW, (input) INTEGER NSUP (input) INTEGER NLOW and NSUP specify the boundary indices for the rows and columns of the principal subpencil of A - lambda*E whose diagonal blocks are to be reordered. 0 <= NLOW <= NSUP <= N, if JOBAE = 'S'. NLOW = MIN( 1, N ), NSUP = N, if JOBAE = 'G'. ALPHA (input) DOUBLE PRECISION The boundary of the domain of interest for the generalized eigenvalues of the pair (A,E). For a continuous-time system (DICO = 'C'), ALPHA is the boundary value for the real parts of the generalized eigenvalues, while for a discrete-time system (DICO = 'D'), ALPHA >= 0 represents the boundary value for the moduli of the generalized eigenvalues. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the state dynamics matrix A. If JOBAE = 'S' then A must be a matrix in real Schur form. On exit, the leading N-by-N part of this array contains the matrix Q'*A*Z in real Schur form, with the elements below the first subdiagonal set to zero. The leading NDIM-by-NDIM part of the principal subpencil A12 - lambda*E12, defined by A12 := A(NLOW:NSUP,NLOW:NSUP) and E12 := E(NLOW:NSUP,NLOW:NSUP), has generalized eigenvalues in the domain of interest, and the trailing part of this subpencil has generalized eigenvalues outside the domain of interest. The domain of interest for eig(A12,E12), the generalized eigenvalues of the pair (A12,E12), is defined by the parameters ALPHA, DICO and STDOM as follows: For DICO = 'C': Real(eig(A12,E12)) < ALPHA if STDOM = 'S'; Real(eig(A12,E12)) > ALPHA if STDOM = 'U'. For DICO = 'D': Abs(eig(A12,E12)) < ALPHA if STDOM = 'S'; Abs(eig(A12,E12)) > ALPHA if STDOM = 'U'. LDA INTEGER The leading dimension of the array A. LDA >= MAX(1,N). E (input/output) DOUBLE PRECISION array, dimension (LDE,N) On entry, the leading N-by-N part of this array must contain the descriptor matrix E. If JOBAE = 'S', then E must be an upper triangular matrix. On exit, the leading N-by-N part of this array contains an upper triangular matrix Q'*E*Z, with the elements below the diagonal set to zero. The leading NDIM-by-NDIM part of the principal subpencil A12 - lambda*E12 (see description of A) has generalized eigenvalues in the domain of interest, and the trailing part of this subpencil has generalized eigenvalues outside the domain of interest. LDE INTEGER The leading dimension of the array E. LDE >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,M) On entry, the leading N-by-M part of this array must contain the input matrix B. On exit, the leading N-by-M part of this array contains the transformed input matrix Q'*B. LDB INTEGER The leading dimension of the array B. LDB >= MAX(1,N). C (input/output) DOUBLE PRECISION array, dimension (LDC,N) On entry, the leading P-by-N part of this array must contain the output matrix C. On exit, the leading P-by-N part of this array contains the transformed output matrix C*Z. LDC INTEGER The leading dimension of the array C. LDC >= MAX(1,P). Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) If COMPQ = 'I': on entry, Q need not be set; on exit, the leading N-by-N part of this array contains the orthogonal matrix Q, where Q' is the product of orthogonal transformations which are applied to A, E, and B on the left. If COMPQ = 'U': on entry, the leading N-by-N part of this array must contain an orthogonal matrix Q1; on exit, the leading N-by-N part of this array contains the orthogonal matrix Q1*Q. LDQ INTEGER The leading dimension of the array Q. LDQ >= MAX(1,N). Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) If COMPZ = 'I': on entry, Z need not be set; on exit, the leading N-by-N part of this array contains the orthogonal matrix Z, which is the product of orthogonal transformations applied to A, E, and C on the right. If COMPZ = 'U': on entry, the leading N-by-N part of this array must contain an orthogonal matrix Z1; on exit, the leading N-by-N part of this array contains the orthogonal matrix Z1*Z. LDZ INTEGER The leading dimension of the array Z. LDZ >= MAX(1,N). NDIM (output) INTEGER The number of generalized eigenvalues of the principal subpencil A12 - lambda*E12 (see description of A) lying inside the domain of interest for eigenvalues. ALPHAR (output) DOUBLE PRECISION array, dimension (N) ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA (output) DOUBLE PRECISION array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, are the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1,...,N, are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.

DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= 8*N+16, if JOBAE = 'G'; LDWORK >= 4*N+16, if JOBAE = 'S'. For optimum performance LDWORK should be larger. If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the DWORK array, returns this value as the first entry of the DWORK array, and no error message related to LDWORK is issued by XERBLA.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: the QZ algorithm failed to compute all generalized eigenvalues of the pair (A,E); = 2: a failure occured during the ordering of the generalized real Schur form of the pair (A,E).

If JOBAE = 'G', the pair (A,E) is reduced to an ordered generalized real Schur form using an orthogonal equivalence transformation A <-- Q'*A*Z and E <-- Q'*E*Z. This transformation is determined so that the leading diagonal blocks of the resulting pair (A,E) have generalized eigenvalues in a suitably defined domain of interest. Then, the transformations are applied to the matrices B and C: B <-- Q'*B and C <-- C*Z. If JOBAE = 'S', then the diagonal blocks of the subpencil A12 - lambda*E12, defined by A12 := A(NLOW:NSUP,NLOW:NSUP) and E12 := E(NLOW:NSUP,NLOW:NSUP), are reordered using orthogonal equivalence transformations, such that the leading blocks have generalized eigenvalues in a suitably defined domain of interest.

3 The algorithm requires about 25N floating point operations.

None

**Program Text**

* TG01PD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER NMAX, MMAX, PMAX PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 ) INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, $ LDE = NMAX, LDQ = NMAX, LDZ = NMAX ) INTEGER LDWORK PARAMETER ( LDWORK = 8*NMAX+16 ) * .. Local Scalars .. CHARACTER*1 COMPQ, COMPZ, DICO, JOBAE, STDOM INTEGER I, INFO, J, M, N, NDIM, NLOW, NSUP, P DOUBLE PRECISION ALPHA, TOL * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX), $ B(LDB,MMAX), BETA(NMAX), C(LDC,NMAX), $ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX), $ Z(LDZ,NMAX) * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL TG01PD * .. Intrinsic Functions .. INTRINSIC DCMPLX * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) N, M, P, DICO, STDOM, JOBAE, COMPQ, COMPZ, $ NLOW, NSUP, ALPHA, TOL IF ( N.LT.0 .OR. N.GT.NMAX ) THEN WRITE ( NOUT, FMT = 99988 ) N ELSE READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N ) READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N ) IF ( M.LT.0 .OR. M.GT.MMAX ) THEN WRITE ( NOUT, FMT = 99987 ) M ELSE READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N ) IF ( P.LT.0 .OR. P.GT.PMAX ) THEN WRITE ( NOUT, FMT = 99986 ) P ELSE READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P ) IF ( LSAME( COMPQ, 'U' ) ) $ READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N ) IF ( LSAME( COMPZ, 'U' ) ) $ READ ( NIN, FMT = * ) ( ( Z(I,J), J = 1,N ), I = 1,N ) * Find the reduced descriptor system * (A-lambda E,B,C). CALL TG01PD( DICO, STDOM, JOBAE, COMPQ, COMPZ, N, M, P, $ NLOW, NSUP, ALPHA, A, LDA, E, LDE, B, LDB, $ C, LDC, Q, LDQ, Z, LDZ, NDIM, ALPHAR, $ ALPHAI, BETA, DWORK, LDWORK, INFO ) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99994 ) NDIM WRITE ( NOUT, FMT = 99997 ) DO 10 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N ) 10 CONTINUE WRITE ( NOUT, FMT = 99996 ) DO 20 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N ) 20 CONTINUE WRITE ( NOUT, FMT = 99993 ) DO 30 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M ) 30 CONTINUE WRITE ( NOUT, FMT = 99992 ) DO 40 I = 1, P WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N ) 40 CONTINUE WRITE ( NOUT, FMT = 99991 ) DO 50 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N ) 50 CONTINUE WRITE ( NOUT, FMT = 99990 ) DO 60 I = 1, N WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N ) 60 CONTINUE WRITE ( NOUT, FMT = 99985 ) DO 70 I = 1, N IF ( BETA(I).EQ.ZERO .OR. ALPHAI(I).EQ.ZERO ) THEN WRITE ( NOUT, FMT = 99984 ) $ ALPHAR(I)/BETA(I) ELSE WRITE ( NOUT, FMT = 99984 ) $ DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I) END IF 70 CONTINUE END IF END IF END IF END IF STOP * 99999 FORMAT (' TG01PD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from TG01PD = ',I2) 99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ') 99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ') 99995 FORMAT (20(1X,F8.4)) 99994 FORMAT (' Number of eigenvalues in the domain =', I5) 99993 FORMAT (/' The transformed input/state matrix Q''*B is ') 99992 FORMAT (/' The transformed state/output matrix C*Z is ') 99991 FORMAT (/' The left transformation matrix Q is ') 99990 FORMAT (/' The right transformation matrix Z is ') 99988 FORMAT (/' N is out of range.',/' N = ',I5) 99987 FORMAT (/' M is out of range.',/' M = ',I5) 99986 FORMAT (/' P is out of range.',/' P = ',I5) 99985 FORMAT (/' The finite generalized eigenvalues are '/ $ ' real part imag part ') 99984 FORMAT (1X,F9.4,SP,F9.4,S,'i ') END

TG01PD EXAMPLE PROGRAM DATA 4 2 2 C S G I I 1 4 -1.E-7 0.0 -1 0 0 3 0 0 1 2 1 1 0 4 0 0 0 0 1 2 0 0 0 1 0 1 3 9 6 3 0 0 2 0 1 0 0 0 0 1 1 1 -1 0 1 0 0 1 -1 1

TG01PD EXAMPLE PROGRAM RESULTS Number of eigenvalues in the domain = 1 The transformed state dynamics matrix Q'*A*Z is -1.6311 2.1641 -3.6829 -0.3369 0.0000 0.4550 -1.9033 0.6425 0.0000 0.0000 2.6950 0.6882 0.0000 0.0000 0.0000 0.0000 The transformed descriptor matrix Q'*E*Z is 0.4484 9.6340 -1.2601 -5.6475 0.0000 3.3099 0.6641 -1.4869 0.0000 0.0000 0.0000 -1.3765 0.0000 0.0000 0.0000 2.0000 The transformed input/state matrix Q'*B is 0.0232 -0.9413 -0.7251 -0.2478 0.6882 -0.2294 1.0000 1.0000 The transformed state/output matrix C*Z is -0.8621 0.3754 0.3405 1.0000 -0.1511 -1.1192 0.8513 -1.0000 The left transformation matrix Q is 0.0232 -0.7251 0.6882 0.0000 -0.3369 0.6425 0.6882 0.0000 -0.9413 -0.2478 -0.2294 0.0000 0.0000 0.0000 0.0000 1.0000 The right transformation matrix Z is 0.8621 -0.3754 -0.3405 0.0000 -0.4258 -0.9008 -0.0851 0.0000 0.0000 0.0000 0.0000 1.0000 0.2748 -0.2184 0.9364 0.0000 The finite generalized eigenvalues are real part imag part -3.6375 0.1375 Infinity 0.0000